Explorations of a family of stochastic Newmark methods in engineering dynamics

A family of stochastic Newmark methods are explored for direct (path-wise or strong) integrations of stochastically driven dynamical systems of engineering interest. The stochastic excitations are assumed to be modeled by white noise processes or their filters and may be applied additively or multiplicatively. The family of stochastic Newmark maps are developed through a two-parameter, implicit Ito–Taylor expansion of the displacement and velocity vectors associated with the governing stochastic differential equations (SDE-s). Detailed estimates of local and global error orders for the response variables are provided in terms of the given time step size, h. While higher order Newmark methods lead to higher accuracies, far less random variables need to be modeled in the lower order methods to make it much more attractive from a computational point of view. For the specific case of a linear dynamical system, the stochastic Newmark map is used to obtain a closed form map for computing the temporal evolution of the response co-variance matrix. A host of numerical illustrations, covering linear and non-linear, single- and multi-degree-of-freedom dynamical systems, are provided to bring out the advantages and possible weaknesses of the methods proposed.